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Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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Conditional distribution for two random variables

I recently came across and exercise from a past exam and I was wondering how to solve it. Two independent random variables $A$ and $B$ are given and they both follow the exponential distribution but ...
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38 views

Find the marginal pdf of $X$ and $Y$

Suppose that $(U,V)$ has the following joint pdf: $$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$ , where $u\geq0$, $v\geq0$, $\theta>0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $...
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32 views

Bivariate Pareto accumulated distribution function from its density function

I am trying to check how a density function of a bivariate Pareto distribution f(x,y) goes to its accumulated function F(x,y): $$ f(x,y) = \frac{a(a+1)}{b_1b_2}\biggl [-1 + \frac x {b_1} + \frac y{...
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48 views

Is it a distribution function?

Consider $F(x,y) = \mathbb{I}[x+y \geq 1] $ . $\mathbb{I} $ denotes the indicator function . So , $F(x,y)$ is 1 iff $x+y \geq 1$ and $0$ otherwise . Can we consider to be a bivariate distribution ...
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58 views

Finding $L_n$ so that $\lim_{n \rightarrow \infty} P(L_n<\rho<1)=\alpha$

Let $(X_1,Y_1),(X_2,Y_2),..,(X_n,Y_n)$ be iid pairs of random variables with $E(X_1)=E(Y_1)$, $\text{Var}(X_1)=\text{Var}(Y_1)=1$,and $\text{cov}(X_1,Y_1)=\rho \in(-1,1)$. Given $\alpha>0$ , obtain ...
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39 views

Derivatives for box integral of a bivariate normal distribution

I'm having quite a bit of trouble trying to understand how to to calculate partial derivatives of a specific function. Suppose I have the standard bivariate normal density function: $$f(x,y,\rho)=\...
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33 views

What is the probability that both roots of the equation Ax2+Bx+C=0 are real?

Let $A, B,$ and $C$ be independent random variables, uniformly distributed over $[0,10], [0,6],$ and $[0,12]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are ...
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Contours in the bivariate graph connect regions of approximately equal popula-tions. Which of the following interpretations is correct?

Contours in the bivariate (weight, height) graph connect regions of approximately equal popula-tions. Which of the following interpretations is correct? I can not understand anything about this ...
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74 views

Find the PDF for $U=\frac{Y_1}{Y_2}$

I need some help on the following problem: Let $Y_1$ and $Y_2$ be two random variables with the following density function:$$f_1(y_1)= \begin{cases} 6y_1(1-y_1), & \text{if } 0\le y_1\le 1 \\...
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Difference between inverse CDFs in a bivariate RV

X and Y are normal distributions with different mean and variance. They are jointly distributed with some correlation $\rho$ (assume they are simply a bivariate normal RV). Let $F_X(x)$ and $F_Y(x)$ ...
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49 views

Spearman's Correlation Coefficient for Bivariate Normal Distribution

Referring to the answer here https://stats.stackexchange.com/a/66617 It is written that $\rho_s(X_1,X_2) = \rho(F_1(X_1),F_2(X_2))$ My Questions are :- Is that forumla correct? Because I am not ...
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17 views

Bound on probability of bi-variate chi-square distribtuion

If I have two random variables $u_1 = x_1^2$ and $u_2 = x_2^2$ where $x_1,x_2 \sim \mathcal{N}(0,1)$ and $E(x_1x_2) = \rho$. The what is the pdf or bound on the pdf of $P(u_1<c,u_2>c)$? Edit: ...
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14 views

Variance of truncated 2d Gaussian

To find the expectation of the Truncated Gaussian E($z_1^2$| $z_1^2 \leq \tau , z_2^2 \geq \tau$). Where $\boldsymbol{z} = [z_1,z_2]^T$ and $\boldsymbol{z} \sim \mathcal{N}(\boldsymbol{0},C)$, where $...
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184 views

MLE of Parameters of Bivariate Normal Distribution

I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function: $f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\...
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22 views

Calculate a 2-dimensional Gauss-Hermite quadrature approximation

I need to calculate an integral in which the second term is a bivariate normal density. I thought about using a 2-dimensional Gauss-Hermite quadrature. But not being familiar with the subject, I do ...
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63 views

Linear combination of normal distribution which is not normal

Let $\xi_1, \xi_2$ be i.i.d N(0,1). Define $(X_1,X_2)=\begin{cases} (\xi_1, |\xi_2|) \quad \xi_1 \geq 0 \\ (\xi_1,-|\xi_2|) \quad \xi_1 < 0 \end{cases}$ This means we can rewrite $X_1=\xi_1$ and ...
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2answers
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Proving $(X,Y)$ is a normal vector when $X\sim N(1,1)$ and $Y\mid X\sim N(3X,4)$

Suppose I have a random vector $(X,Y)$ with $X\sim\mathcal{N}(1,1)$ and $Y|X = x \sim\mathcal{N}(3x,4)$. I need to prove that $(X,Y)$ is a normal vector as well. To do that I want to explicitly ...
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Find Distribution of Sum of Random Variables given only Joint Distribution

So if we have two random variables $X, Y$, with unknown distributions. Then we have a random variable $Z$, such that $Z=X+Y$. Firstly, how do we find the CDF of $Z$, i.e. $F_Z(z)$, given the joint ...
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1answer
36 views

Correlation Coefficient r, Formula Explained Intuitively

I've seen several videos, Khan Academy included, explaining the correlation coefficient formula but none explain the "logic" behind the formula, not to my satisfaction anyways. The Formula: ...
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27 views

Expectation computation of correlated normal random variables

I'm struggling to prove the following maybe some of you can help me. Here is my problem Let $\mathbf{X}\sim\mathcal{N}(\mu,\Sigma)$ where $\mathbf{X}=\pmatrix{X_1\\X_2}, \mu=\pmatrix{\mu_1\\\mu_2}$ ...
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1answer
38 views

Find the marginal distribution of an point randomly chosen on an ellipse

This exercise comes from rice 3.6 and states: A point is chosen randomly in the interior of an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Find the marginal densities of the $x$ and $y$ ...
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28 views

Bivariate normal: Expected value and variance

I have this simple exercise to do and I'm new to this topic. Seeing the slides of my professor, I would solve the problem in this way: $E(Y)=c_1 μ_1+c_2 μ_2$ $E(Y)=-54.2424$ $Var(Y)=σ_{22}$ $Var(...
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103 views

Find the expected value and standard deviation of the returns for the following portfolio

Suppose you want a portfolio composed of AT&T, Cigna, Disney, and Ford. Find the expected value and standard deviation of the returns for the following portfolio I know that I have to use ...
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1answer
20 views

Bivariate Normal Proof

In our discussion of the bivariate normal, there is an expression for $E(Y|X = x)$. a. By reversing the roles of X and Y give a similar formula for $E(X|Y = y)$. b. Both $E(Y|X = x)$ and $E(X|Y = y) ...
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83 views

Conditional marginal distribution of conditional bivariate normal distribution

I have a bivariate normal distribution$$(X, Y)\sim N(\mu_{x}, \mu_{y}, \sigma_{x}^2, \sigma_{y}^2, \rho)$$ My question is : when $X > k$ ($k$ is a constant),how to get the distribution of $Y$? Can ...
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conditional and marginal of bivariate t distribution - i.i.d or not?

I have a bi variate t-distribution with $X=(X_1, X_2)=t(\mu, \sigma^2, v)$ $$\mu=(1,2)',\sigma^2=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right), v=4 $$ How can I calculate the ...
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1answer
50 views

Finding the Probability for a Bivariate Normal Distribution [duplicate]

Given $(x_1,x_2)' \sim N_2 \left(\bf{0},\Sigma\right) = N_2 \left(\left(\begin{array}{l}0\\0\end{array}\right), \left(\begin{array}{l}1&\rho\\\rho&1\end{array}\right)\right)$, find $Pr(x_1>...
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37 views

Calculate the joint distribution of $X, Y$ and $Z$, following a bivariate normal distribution

Let $X$ and $Y$ follow a bivariate normal distribution with $\mu = (0,0)^T$, $\sigma_x=1$, $\sigma_y=1$ and correlation $\rho =0.5$. Also, suppose that a pair $Z$ and $Y$ follow a bivariate normal ...
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78 views

Find the distribution of $Y-X$. [closed]

Let $X \sim N(0,1)$ and $Y \sim N(0,2)$ respectively, independently of each other. Find the distribution of $Y-X$. How do I proceed? As I am a beginner I don't know how to approach. Please help me in ...
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Formula for copulas of bivariate mixed (discrete and continuous) data

I have a question as I am new to copulas. I have bivariate data (X,Y), one is discrete and one is continuous distributed. Can I use the usual formula for the common density function given by $$f_X(x) \...
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Is taking $3\sigma$ limits for bivariate correlated random variables as the confidence limits an under estimation?

If the two random variables are uncorrelated, $3\sigma$ limit would mean that the random variables lie inside a hyperrectangle with center coordinates being means of random variables. But if two ...
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Bivariate probability distribution of a pdf

Given $$f ( x_1 , x_2 ) = \begin{cases}6 ( 1 - x_2 ) &,&0 ≤ x_1 ≤ x_2 ≤ 1 \\ 0 &&\text{ Otherwise.}\end{cases}$$ Find $P ( X_1 ≤ 0.75, X_2 ≥ 0.5 )$ The correct answer is $31/64$. ...
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Circle-shaped integration of a bivariate normal distribution

I have a bivariate Normal Distribution of positions in a 2D space given by the marginal distributions of x and y components: $N_x(\mu_x,\sigma_x)$, $N_y(\mu_y,\sigma_y)$. I want to determine the ...
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1answer
18 views

Checking the validity of Bivariate Distribution function

Which of the following is /are bivariate Distribution function: $F(x,y)=\begin{cases} 0 & \text{ if } x+y<0\\ 1 & \text{ if } x+y\geq0 \end{cases}$ $F(x,y)=\begin{cases} 1-e^{-...
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2answers
40 views

calculate $\operatorname{cov}(X,Y)$ from $f_{x,y}(x,y)$

I have the following density function: $$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$ We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$...
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1answer
24 views

Calculate probability $P(X_1\geq1/2, X_2\geq1/2)$ for $F_{X_1,X_2}$

I am studying for an exam and came across this example to calculate probability $P(X_1\geq1/2, X_2\geq1/2)$ for $F_{X_1,X_2}$ = $1/2x_1x_2(x_2^2+x_1^2)$ if $0\leq x_1\leq 1,0\leq x_2\leq 1$. The ...
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1answer
120 views

Distribution of $Z=\frac{UX+VY}{\sqrt{U^2+2\rho UV+V^2}}$ when $(X,Y)$ is bivariate normal and $U,V$ independent of $X$ and $Y$

$X$ and $Y$ have bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Let $U$ and $V$ be independent of $X$ and $Y$. How can we find the distribution of $Z = \...
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0answers
38 views

Conditional normality implies marginal normality or bivariate normality? [closed]

Let X and Y be two random variables. If we know that both Y|X and X|Y are normal, then can we conclude that (1) X and Y are normal respectively; (2) X and Y are bivariate normal? Thanks! Edited: ...
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1answer
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Formula for expectation of Bivariate Data [closed]

`Suppose that $(X, Y )$ has a uniform distribution on the parallelogram with vertices at $(0,0)$ $(1292,1000)$ $(1526,0)$ $(2818,1000)$ Calculate the means of $X$ and of $Y$. I don't know the ...
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1answer
22 views

Bivariate distribution of the discrete type question

A homework questions asks: John has two quarters and two dimes. He tosses these four coins 3 times. Let U be the number of heads of quarters, and V be the number of heads of dimes. Denote X = U +...
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2answers
51 views

Non-normal Bivariate distribution with normal margins

I was working on a question and it stated the following: Now I am supposed to show that the bivariate distribution is non normal even though the marginal distributions are. I generated 1000 iid ...
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1answer
49 views

Joint normal and exponential

if X has exponential exp(lambda)and Y has normal distribution N(0,1) . X and Y are independent. how can one find p(X < Y)? My thoughts were to integrate : $$\int_{-\infty}^{+\infty} F_X(y)f_Y(y) \...
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1answer
273 views

Are uncorrelated linear combinations of the elements of a multivariate normal distribution always independent of each other?

To make a simple example, we could have $X \sim N_2 \left( \begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 4 & -4 \\ -4 & 2\end{pmatrix} \right)$ which gives $X_1 \sim N(0, 4)$ and $X_2 \...
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0answers
52 views

Is the joint distribution of any two uncorrelated normally distributed random variables bivariate normal?

Originally I was only going to ask if the joint distribution of any two normally distributed is random variables bivariate normal, however it seems that a large set of counterexamples can be found ...
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1answer
29 views

Covariance matrix of $(\bar{X}, \bar{X^2})$

I found a pdf that claims that asymptotically: $(\frac{\sum X_i}{N}, \frac{\sum X_i^{2}}{N})$ converges in distribution to a bivariate random variable with mean $(\mu_1,\mu_2)$ and covariance matrix $...
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1answer
500 views

Expectation for Trinomial distribution

I am trying to understand the proof of $\mathrm{E}[xy]=n(n-1)p_1p_2$ where x,y have a trinomial distribution with pmf: $p(x,y) = \frac{n!}{x!y!(n−x−y)!}p_1^xp_2^yp_3^{n−x−y}$ The proof has the ...
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0answers
33 views

What is wrong with the following approach of obtaining the joint distribution function.

Let the joint probability density function of $X,Y$ is given as: $f(x,y)=\frac{1}{4}[1+x^3y^3]$ where $-1\leq x \leq 1$ and $-1\leq y \leq 1$. It is required to obtain the joint probability density ...
6
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1answer
318 views

Variance of Z = max(X,Y) where X Y are jointly bivariate normal

I have a question about the bivariate normal r.v.'s Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$. My ...
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2answers
58 views

Find the density function of the sum $(X,X+Y)$.

Problem I have to prove the following: Let $X$ and $Y$ be independent continuous random variables with density function $f:\mathbb R\to\mathbb R$. \ Prove that $(X,X+Y)$ is a continuous bivariate ...
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1answer
720 views

Finding the conditional probability given the joint probability density function

f(x,y) = e^-x if 0 <= y <=x < \infty i need to find the P(X<3|Y<2) & P(X<3|Y=2). i'm struggling with the first probability. i'm not sure how to evaluate the conditional given ...